Transcript Molekylfysik - Leiden Univ
= c 1. Quantum theory: introduction and principles
1.1 The failures of classical physics 1.2 Wave-particle duality 1.3 The Schrödinger equation 1.4 The Born interpretation of the wavefunction 1.5 Operators and theorems of the quantum theory 1.6 The Uncertainty Principle
1.1 The failures of classical physics A. Black-body radiation
at
max
Observations: 1)
Wien displacement law
: 2)
Stefan-Boltzmann law
: (
= E/V)
Tentative explanation via the classical mechanics
Equipartition of the energy
: Per degree of freedom: average energy = kT (26 meV at 25°C).
The total energy is equally “partitioned” over all the available modes of motion.
Rayleigh and Jeans used the equipartition principle and consider that the electromagnetic radiation is emitted from a collection of excited oscillators, which can have
any given energy
by controlling the applied forces (related to T). It led to the
Rayleigh-Jeans law
function of the wavelength .
for the energy density as a It does not fit to the experiment. From this law, every objects should emit IR, Vis, UV, X-ray radiation. There should be no darkness!! This is called the
Ultraviolet catastrophe
.
Introduction of the quantization of energy to solve the black-body problem
Max Planck:
quantization of energy
.
E = n
h
only for n= 0,1,2, ...
h
is the Planck constant
Cross-check of the theory
: from the Planck distribution, one can easily find the experimental Wien displacement and the Stefan-Boltzmann law.
the quantization of energy exists!
C. Atomic and molecular spectra
Excitation energy
Fe
Photon emission h =hc/ The
emission and absorption
of radiation always occurs at
specific frequencies
: another proof of the energy quantization.
Photon absorption NB: wavenumber ~
c
1
1.2 Wave-particle duality
A. The particle character of electromagnetic radiation:
The photoelectric effect
h
e
-
(E
k
) metal
The
photon h
↔
particle-like projectile
conservation of energy ½mv 2 = h
-
=
metal workfunction
, the minimum energy required to remove an electron from the metal to the infinity.
Threshold does not depend on intensity of incident radiation.
NB: The
photoelectron spectroscopy
this photoelectric effect .
(UPS, XPS) is based on
B. The wave character of the particles:
Electron diffraction Diffraction is a characteristic property of waves. With X-ray, Bragg showed that a constructive interference occurs when =2d sin . Davidsson and Germer showed also interference phenomenon but with electrons!
Particles are characterized by a wavefunction
A link between the particle (p=mv) and the wave ( ) natures
V
An appropriate potential difference creates electrons that can diffract with the lattice of the nickel
d
1.3 The Schrödinger Equation
From the wave-particle duality, the concepts of classical physics (CP) have to be abandoned to describe microscopic systems. The
dynamics of microscopic
be described in a new theory:
the quantum theory (QT)
.
systems will A wave, called
wavefunction
(r,t)
,
is associated to each object. The well-defined trajectory of an object in CP (the location, r, and momenta, p = m.v, are precisely known at each instant t) is replaced by
(r,t)
indicating that the particle is distributed through space like a wave. In QT, the location, r, and momenta, p, are not precisely known at each instant t (see
Uncertainty Principle
).
In CP, all modes of motions (rot, trans, vib) can have any given energy by controlling the applied forces. In the QT, all modes of motion cannot have any given energy, but can only be excited at
specified energy levels
(see
quantization of energy
).
The Planck constant
h
can be a
criterion
to know if a problem has to be addressed in CP or in QT.
ML 2 T -1
(E= h
h
can be seen has a where E is in ML 2 T “
quantum of an action
” -2 and is in T -1 that has the dimension of ). With the specific parameters of a problem, we built a quantity having the dimension of an action (ML 2 T -1 ). If this quantity has the order of magnitude of
h
( 10 -34 Js), the problem has to be treated within the QT.
Classical mechanics
H H
Hamiltonian function H = T + V
.
T is the kinetic energy and V is the potential energy.
correspondence principles
are proposed to pass from the classical mechanics to the quantum mechanics
T
V
p
2 2
m
E
V
(
x
,
y
,
z
,
t
)
x
x
p
i grad
i
x
y
z
p
2 2 2 2
x
2 2
E
i
t V
(
x
,
y
,
z
,
t
)
V
(
x
,
y
,
z
,
t
)
y
2 2
z
2 2 Quantum mechanics
H
i
t
Schrödinger Equation
H
2 2
m
2
V
(
x
,
y
,
z
,
t
)
The
Schrödinger Equation
(SE) shows that the operator H and iħ / t give the same results when they act on the wavefunction. Both are equivalent operators corresponding to the total energy E.
H H
i
t
2 2
m
2
V
(
x
,
y
,
z
,
t
) In the case of
stationary systems
, the potential V(x,y,z) is wavefunction can be written as a stationary wave: (x,y,z,t)=
time independent
. The (x,y,z) e -i t (with E=ħ ).
This solution of the SE leads to a density of probability | (x,y,z,t)| 2 = | (x,y,z)| 2 , which is independent of time. The
Time Independent Schrödinger Equation
is: 2 2
m
2
V
(
x
,
y
,
z
) (
x
,
y
,
z
)
E
(
x
,
y
,
z
) or
H
E
NB: In the following, we only envisage the time independent version of the SE.
The Schrödinger equation is an
eigenvalue equation
, which has the typical form:
(operator)(function)=(constant)×(same function)
The
eigenvalue
is the energy E. The set of eigenvalues are the only values that the energy can have (quantization).
The system.
eigenfunctions
of the Hamiltonian operator H are the wavefunctions of the To each eigenvalue corresponds a set of eigenfunctions. Among those, only the eigenfunctions that fulfill specific conditions have a physical meaning.
1.4 The Born interpretation of the wavefunction
Physical meaning of the wavefunction
:
probability of finding
the particle in an infinitesimal volume d =dxdydz at some point
r
is proportional to
|
(r)| 2 d
Example of a 1-dimensional system | (r)| 2 = (r) * (r) is a
probability density
.
It is always positive!
wavefunction may have negative or complex values Node
A. Normalization Condition
The solution of the differential equation of Schrödinger is defined
within a constant N
. If is a known solution of H =E , then = N is a also solution for the same E.
H =E H(N )= E(N ) N(H )=N(E ) H =E T he sum of the probability of finding the particle over all infinitesimal volumes d of the space is 1 :
Normalization condition
. We have to determine the constant N, such that the solution =N of the SE is normalized.
*
d
1 (
N
' )(
N
' * )
d
1
N
2 ' ' *
d
1
N
1 ' ' *
d
B. Other mathematical conditions
(r) ; r (r) should be single-valued r if not: 2 probability for the same point!!
*
d
???
The SE is a second-order differential equation: (r) and d (r)/dr should be continuous
C. The kinetic energy and the wavefunction
T
H
T
V
2 2
m
x
2 2
V
T
* 2 2
m
x
2 2
d
2 2
m
* 2
x
2
d
We can expect a particle to have a high kinetic energy if the average curvature of its wavefunction is high.
The kinetic energy is then a kind of average over the curvature of the wavefunction: we get a large contribution to the observed value from the regions where the wavefunction is sharply curved (
2
/
x 2
is large) and the wavefunction itself is large ( * is large too).
Example: the wave function in a periodic system: electrons in a metal
1.5 Operators and principles of quantum mechanics
A. Operators in the quantum theory (QT)
An eigenvalue equation,
f
=
f,
operators are linear and hermitian.
can be associated to each operator
.
In the QT, the
Linearity:
is linear if: (c
f)= c
f
(c=constant) and NB: “c” can be defined to fulfill the normalization condition (
f+
)
=
f+
Hermiticity:
A linear operator is hermitian if:
f
*
d
*
f
*
d
where
f
and are finite, uniform, continuous and the integral for the normalization converge.
The eigenvalues of an hermitian operator are real numbers ( = * ) When the operator of an eigenvalue equation is hermitian, corresponding to 2 different eigenvalues (
j
,
k
) are orthogonal.
f j
f k
j
k f j f k
f j f k
*
d
0 2 eigenfunctions (
f j , f k
)
B. Principles of Quantum mechanics
1. To each
and hermitian
eigenvalues
j
observable or measurable property
operator < > of the system corresponds a
linear
, such that the only measurable values of this observable are the of the corresponding operator.
f
=
f
2. Each
hermitian operator
representing a physical property is “
complete
”.
Def: An operator is “
complete
” if any function (finite, uniform and continuous) can be developed as a series of eigenfunctions
f j
of this operator.
(x,y,z) (
x
,
y
,
z
)
j C j f j
(
x
,
y
,
z
) 3. If (x,y,z) is a solution of the Schrödinger equation for a particle, and if we want to measure the value of the observable related to the complete and hermitian operator (that is not the Hamiltonian), then the
probability to measure the eigenvalue
k
is equal to the square of the modulus of
f k
’s coefficient, that is
|C k | 2
, for an othornomal set of eigenfunctions {
f j
}.
Def: The eigenfunctions are orthonormal if
f j f k
*
d
ij
NB: In this case:
j C j
2 1
4. The average value of a large number of observations is given by < > of the operator an operator corresponding to the observable of interest. The expectation value of is defined as: the
expectation value
* *
d
d
For normalized wavefunction *
d
j C j
2
j
See p305 in the book 5.
If the wavefunction expectation value of
=f
1
is the eigenfunction of the operator is the eigenvalue 1 .
(
f
=
f
), then the *
d
* 1
d
1 *
d
1
1.6 The Uncertainty Principle
1. When two operators are
commutable
(and with the Hamiltonian operator), their eigenfunctions are common and the corresponding observables can be
determined simultaneously and accurately .
2. Reciprocally, if two operators
do not commute
, the corresponding observable
be determined simultaneously and accurately.
cannot
If ( 1 2 2 1 ) = c, where “c” is a constant, then an the measurement of these two observables: uncertainty relation takes place for where 1 1 2 1 2 1 / 2 1 2
c
2 Uncertainty Principle
Example of the Uncertainty Principle
1. For a free atom and without taking into account the spin-orbit coupling, the angular orbital moment L 2 and the total spin S 2 commute with the Hamiltonian H. Hence, an exact value of the eigenvalues L of L good quantum numbers 2 and S of S 2 can be measured simultaneously.
to characterize the wavefunction of a free atom L and S are see Chap 13 “Atomic structure and atomic spectra”.
2.
The position x and the momentum p x
(along the x axis). According to the correspondence principles, the quantum operators are:
x
and
ħ/i(
/
x)
. The commutator can be calculated to be:
i
x
,
x
i
p x
x
2 The consequence is a breakdown of the classical mechanics laws: if there is a complete certainty about the position of the particle ( x=0), then there is a complete uncertainty about the momentum ( p x = ).
3.
The time and the energy
:
t
E
If a system stays in a state during a time 2 t, the energy of this system cannot be determined more accurately than with an error E.
This incertitude is of major importance for all spectroscopies : see Chap 16, 17, 18